A real-tree is a metric space X such that between any two points in X there is a unique arc, and that arc is a geodesic segment. This thesis shows that the class of pointed, complete real-trees is axiomatizable in a suitable continuous signature. The thesis then describes a model companion of the theory of real-trees that has quantifier elimination, is complete and is stable but not superstable. The model theoretic independence relation for the model companion is described and it is shown that it is not categorical in any infinite cardinal. Next it is shown that various classes of pointed, complete real-trees with isometries are axiomatizable, and model companions are found for those theories. The thesis characterizes the completions of the model companions and shows they are stable but not superstable, describes their independence relations and shows they are not categorical in any infinite cardinal.